Efficient, sparse representation of manifold distance matrices for classical scaling

Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points. Thus for large point sets it is common to use a low-rank approximation to the distance matrix, which fits in memory and can be efficiently analyzed using methods such as multidimensional scaling (MDS). In this paper we present a novel sparse method for efficiently representing geodesic distance matrices using biharmonic interpolation. This method exploits knowledge of the data manifold to learn a sparse interpolation operator that approximates distances using a subset of points. We show that our method is 2x faster and uses 20x less memory than current leading methods for solving MDS on large point sets, with similar quality. This enables analyses of large point sets that were previously infeasible.Comment: Conference CVPR 201

On the stability of circular orbits in galactic dynamics: Newtonian thin disks

The study of off-equatorial orbits in razor-thin disks is still in its beginnings. Contrary to what was presented in the literature in recent publications, the vertical stability criterion for equatorial circular orbits cannot be based on the vertical epicyclic frequency, because of the discontinuity in the gravitational field on the equatorial plane. We present a rigorous criterion for the vertical stability of circular orbits in systems composed by a razor-thin disk surrounded by a smooth axially symmetric distribution of matter, the latter representing additional structures such as thick disk, bulge and (dark matter) halo. This criterion is satisfied once the mass surface density of the thin disk is positive. Qualitative and quantitative analyses of nearly equatorial orbits are presented. In particular, the analysis of nearly equatorial orbits allows us to construct an approximate analytical third integral of motion in this region of phase-space, which describes the shape of these orbits in the meridional plane.Comment: 3 pages, 1 figure. In Proceedings of the MG13 Meeting on General Relativity, Stockholm University, Sweden, 1-7 July 2012. World Scientific, Singapore. Based on arXiv:1206.6501. in The Thirteenth Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (In 3 Volumes), chap. 438, pages 2346-2348 (2015

Economic Inequality in Spain: The European Union Household Panel Dataset

This article uses data from the 1998 European Union Household Panel to study economic inequality in Spain. It reports data on the Spanish distributions of income, labor income, and capital income, and on related features of inequality, such as age, employment status, educational attainment, and marital status. It also reports data on the income mobility of Spanish households. We find that income, earnings, and, very especially, capital income are very unequally distributed in Spain

Vertical stability of circular orbits in relativistic razor-thin disks

During the last few decades, there has been a growing interest in exact solutions of Einstein equations describing razor-thin disks. Despite the progress in the area, the analytical study of geodesic motion crossing the disk plane in these systems is not yet so developed. In the present work, we propose a definite vertical stability criterion for circular equatorial timelike geodesics in static, axially symmetric thin disks, possibly surrounded by other structures preserving axial symmetry. It turns out that the strong energy condition for the disk stress-energy content is sufficient for vertical stability of these orbits. Moreover, adiabatic invariance of the vertical action variable gives us an approximate third integral of motion for oblique orbits which deviate slightly from the equatorial plane. Such new approximate third integral certainly points to a better understanding of the analytical properties of these orbits. The results presented here, derived for static spacetimes, may be a starting point to study the motion around rotating, stationary razor-thin disks. Our results also allow us to conjecture that the strong energy condition should be sufficient to assure transversal stability of periodic orbits for any singular timelike hypersurface, provided it is invariant under the geodesic flow.Comment: 13 pages, 4 figures; Accepted for publication in Physical Review